/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 257 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) TRS for Loop Detection


----------------------------------------

(0)
Obligation:
The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   a__f(X) -> a__if(mark(X), c, f(true))
   a__if(true, X, Y) -> mark(X)
   a__if(false, X, Y) -> mark(Y)
   mark(f(X)) -> a__f(mark(X))
   mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
   mark(c) -> c
   mark(true) -> true
   mark(false) -> false
   a__f(X) -> f(X)
   a__if(X1, X2, X3) -> if(X1, X2, X3)

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(c) -> c
   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1) -> a__f(encArg(x_1))
   encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_c -> c
   encode_f(x_1) -> f(encArg(x_1))
   encode_true -> true
   encode_false -> false
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))


----------------------------------------

(2)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   a__f(X) -> a__if(mark(X), c, f(true))
   a__if(true, X, Y) -> mark(X)
   a__if(false, X, Y) -> mark(Y)
   mark(f(X)) -> a__f(mark(X))
   mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
   mark(c) -> c
   mark(true) -> true
   mark(false) -> false
   a__f(X) -> f(X)
   a__if(X1, X2, X3) -> if(X1, X2, X3)

The (relative) TRS S consists of the following rules:

   encArg(c) -> c
   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1) -> a__f(encArg(x_1))
   encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_c -> c
   encode_f(x_1) -> f(encArg(x_1))
   encode_true -> true
   encode_false -> false
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

Rewrite Strategy: FULL
----------------------------------------

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   a__f(X) -> a__if(mark(X), c, f(true))
   a__if(true, X, Y) -> mark(X)
   a__if(false, X, Y) -> mark(Y)
   mark(f(X)) -> a__f(mark(X))
   mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
   mark(c) -> c
   mark(true) -> true
   mark(false) -> false
   a__f(X) -> f(X)
   a__if(X1, X2, X3) -> if(X1, X2, X3)

The (relative) TRS S consists of the following rules:

   encArg(c) -> c
   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1) -> a__f(encArg(x_1))
   encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_c -> c
   encode_f(x_1) -> f(encArg(x_1))
   encode_true -> true
   encode_false -> false
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

Rewrite Strategy: FULL
----------------------------------------

(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(6)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   a__f(X) -> a__if(mark(X), c, f(true))
   a__if(true, X, Y) -> mark(X)
   a__if(false, X, Y) -> mark(Y)
   mark(f(X)) -> a__f(mark(X))
   mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
   mark(c) -> c
   mark(true) -> true
   mark(false) -> false
   a__f(X) -> f(X)
   a__if(X1, X2, X3) -> if(X1, X2, X3)

The (relative) TRS S consists of the following rules:

   encArg(c) -> c
   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1) -> a__f(encArg(x_1))
   encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_c -> c
   encode_f(x_1) -> f(encArg(x_1))
   encode_true -> true
   encode_false -> false
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

Rewrite Strategy: FULL
----------------------------------------

(7) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

mark(f(X)) ->^+ a__f(mark(X))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [X / f(X)].

The result substitution is [ ].




----------------------------------------

(8)
Complex Obligation (BEST)

----------------------------------------

(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   a__f(X) -> a__if(mark(X), c, f(true))
   a__if(true, X, Y) -> mark(X)
   a__if(false, X, Y) -> mark(Y)
   mark(f(X)) -> a__f(mark(X))
   mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
   mark(c) -> c
   mark(true) -> true
   mark(false) -> false
   a__f(X) -> f(X)
   a__if(X1, X2, X3) -> if(X1, X2, X3)

The (relative) TRS S consists of the following rules:

   encArg(c) -> c
   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1) -> a__f(encArg(x_1))
   encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_c -> c
   encode_f(x_1) -> f(encArg(x_1))
   encode_true -> true
   encode_false -> false
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

Rewrite Strategy: FULL
----------------------------------------

(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(11)
BOUNDS(n^1, INF)

----------------------------------------

(12)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   a__f(X) -> a__if(mark(X), c, f(true))
   a__if(true, X, Y) -> mark(X)
   a__if(false, X, Y) -> mark(Y)
   mark(f(X)) -> a__f(mark(X))
   mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
   mark(c) -> c
   mark(true) -> true
   mark(false) -> false
   a__f(X) -> f(X)
   a__if(X1, X2, X3) -> if(X1, X2, X3)

The (relative) TRS S consists of the following rules:

   encArg(c) -> c
   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1) -> a__f(encArg(x_1))
   encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_c -> c
   encode_f(x_1) -> f(encArg(x_1))
   encode_true -> true
   encode_false -> false
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

Rewrite Strategy: FULL